Observables as Hermitian Operators and Their Spectra
Observables are represented by self-adjoint operators, or Hermitian matrices in finite dimensions. Their spectra contain the possible measurement values. Discrete outcomes use eigenspaces; continuous outcomes use spectral densities and generalised eigenstates.
What a learner can do afterwards
- Shows that a Hermitian matrix has real eigenvalues and that eigenvectors with distinct eigenvalues are orthogonal
- Expands a state in a discrete orthonormal eigenbasis and predicts outcome probabilities, combining weights for a degenerate eigenvalue
- Explains why a continuous spectrum forces a different normalisation convention
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An observable is represented by a self-adjoint operator. In finite dimensions this means a Hermitian matrix A = A†, where † is conjugate transpose. Such a matrix has real eigenvalues and an orthonormal eigenbasis. Real eigenvalues alone are not enough to make a matrix Hermitian. For unbounded operators, the domain is also part of the self-adjointness condition.
<u|A|v> means the inner product of u with Av. Hermiticity gives <Au|v> = <u|Av>, making <v|A|v> equal its conjugate. For Av = av, v ≠ 0, a = <v|A|v>/<v|v> is real. If Au = au and Av = bv, then (a - b)<u|v> = 0: distinct eigenvalues imply orthogonality. Within a degenerate eigenspace, an orthonormal basis can be chosen; arbitrary eigenvectors need not be orthogonal.
For a discrete orthonormal eigenbasis, write |ψ> = Σ cⱼ|j>. A normalised state has Σ |cⱼ|² = 1. For two distinct eigenvalues and coefficients 1/√5 and 2/√5, the probabilities are 1/5 and 4/5. If several basis vectors share one eigenvalue, add their squared coefficient magnitudes to get that outcome's probability.
An observable's spectrum is its set of possible measurement values. It may contain discrete eigenvalues, continuous parts, or both. Some spectral values have no normalisable eigenvectors, so the spectrum need not equal the ordinary eigenvalue set. Hydrogen illustrates both kinds: discrete bound energies and a continuous ionisation spectrum.
An atomic emission-line photon carries an energy difference: E(photon) = E(upper) - E(lower) = hf. The line therefore indicates a transition between energy levels, not the absolute value of one energy eigenvalue. Absorption reverses the energy transfer. Finite lifetimes and experimental effects give real lines nonzero widths.
An exact momentum eigenfunction on the full real line is a plane wave with constant magnitude, so its squared magnitude has an infinite integral. Its generalised ket uses delta normalisation, <p|p'> = δ(p - p'), not ordinary unit norm. A physical wave packet instead has ∫ |φ(p)|² dp = 1. Integrating this density over a momentum interval gives its probability.
For a normalised state in a discrete orthonormal eigenbasis, sum squared coefficient magnitudes for each eigenvalue. Continuous probabilities use density integrals; self-adjoint spectra are real.
Lesson and questions written for this dataset using standard results for Hermitian matrices and quantum observables.
Readings written for this dataset. The flashcards were drafted from collected fact candidates whose individual sources were not recorded.
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